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Theorem ballotfilemirc 13219

Description: Applying 𝑅 does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.) (Revised by AV, 6-Oct-2020.)

**Hypotheses**
Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfilem.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfilem.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))

**Assertion**
Ref	Expression
ballotfilemirc	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐶))

Distinct variable groups: 𝑀,𝑐 𝑁,𝑐 𝑂,𝑐 𝑖,𝑀 𝑖,𝑁 𝑖,𝑂 𝑘,𝑀 𝑘,𝑁 𝑘,𝑂 𝑖,𝑐,𝐹,𝑘 𝐶,𝑖,𝑘 𝑖,𝐸,𝑘 𝐶,𝑘 𝑘,𝐼,𝑐 𝐸,𝑐 𝑖,𝐼,𝑐 𝑆,𝑘,𝑖,𝑐 𝑅,𝑖,𝑘 𝑥,𝑘,𝐶 𝑥,𝐹 𝑥,𝑀 𝑥,𝑁 Allowed substitution hints: 𝐶(𝑐) 𝑃(𝑥,𝑖,𝑘,𝑐) 𝑅(𝑥,𝑐) 𝑆(𝑥) 𝐸(𝑥) 𝐼(𝑥) 𝑂(𝑥)

Proof of Theorem ballotfilemirc Dummy variables 𝑦 𝑣 𝑢 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . 4 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . 4 ⊢ 𝑁 ∈ ℕ
3		ballotfilem.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
4		ballotfilem.p	. . . 4 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . . 4 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . 4 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . 4 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . 4 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9		ballotth.s	. . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10		ballotth.r	. . . 4 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemrc 13218	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸))
12	1, 2, 3, 4, 5, 6, 7, 8	ballotfilemi 13187	. . 3 ⊢ ((𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < ))
13	11, 12	syl 14	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < ))
14		lttri3 8369	. . . 4 ⊢ ((𝑝 ∈ ℝ ∧ 𝑞 ∈ ℝ) → (𝑝 = 𝑞 ↔ (¬ 𝑝 < 𝑞 ∧ ¬ 𝑞 < 𝑝)))
15	14	adantl 277	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑝 ∈ ℝ ∧ 𝑞 ∈ ℝ)) → (𝑝 = 𝑞 ↔ (¬ 𝑝 < 𝑞 ∧ ¬ 𝑞 < 𝑝)))
16	1, 2, 3, 4, 5, 6, 7, 8	ballotfilemiex 13188	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
17	16	simpld 112	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
18	17	elfzelzd 10379	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
19	18	zred 9718	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℝ)
20		fveqeq2 5684	. . . 4 ⊢ (𝑘 = (𝐼‘𝐶) → (((𝐹‘(𝑅‘𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0))
21		eqid 2234	. . . . 5 ⊢ (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21	ballotfilemfrci 13215	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0)
23	20, 17, 22	elrabd 2978	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0})
24		elrabi 2973	. . . . 5 ⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → 𝑦 ∈ (1...(𝑀 + 𝑁)))
25	24	anim2i 342	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → (𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))))
26	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemfrcn0 13217	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝑦) ≠ 0)
27	26	neneqd 2435	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0)
28		fveqeq2 5684	. . . . . . . . . 10 ⊢ (𝑘 = 𝑦 → (((𝐹‘(𝑅‘𝐶))‘𝑘) = 0 ↔ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0))
29	28	elrab 2976	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} ↔ (𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0))
30	29	simprbi 275	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → ((𝐹‘(𝑅‘𝐶))‘𝑦) = 0)
31	27, 30	nsyl 633	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 < (𝐼‘𝐶)) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0})
32	31	3expia 1232	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 < (𝐼‘𝐶) → ¬ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}))
33	32	con2d 629	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → (𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0} → ¬ 𝑦 < (𝐼‘𝐶)))
34	33	imp 124	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼‘𝐶))
35	25, 34	sylancom 420	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}) → ¬ 𝑦 < (𝐼‘𝐶))
36	15, 19, 23, 35	infminti 7331	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘(𝑅‘𝐶))‘𝑘) = 0}, ℝ, < ) = (𝐼‘𝐶))
37	13, 36	eqtrd 2267	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐶))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ∀wral 2522 {crab 2526 ∖ cdif 3211 ∩ cin 3213 ifcif 3624 𝒫 cpw 3674 class class class wbr 4114 ↦ cmpt 4176 “ cima 4757 ‘cfv 5357 (class class class)co 6058 ∈ cmpo 6060 Fincfn 6988 infcinf 7287 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 < clt 8324 ≤ cle 8325 − cmin 8460 / cdiv 8963 ℕcn 9254 ℤcz 9594 ...cfz 10361 ♯chash 11163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-fz 10362 df-fzo 10499 df-ihash 11164

This theorem is referenced by: ballotfilemrinv0 13220

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